How this calculator works
Magnitude is the distance from the origin in the complex plane. The two-argument arctangent uses both component signs to choose the correct quadrant, including points on the axes. A whole turn can then be added to express the same point in the requested interval. Phase readouts support angular units; the exponential text and step table explicitly use radians. At zero there is no direction, so phase is omitted.
Formula
r=√(a²+b²); θ=atan2(b,a); a+bi = r exp(iθ). Equivalent arguments are θ+2πk for integers k.
Worked example
For -1+i, r=√2 and θ=135°. For -1-i the signed phase is -135°, while the positive-interval phase is 225°. Both identify the same third-quadrant point.
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Frequently asked questions
Why not simply use arctan(b/a)?
That quotient loses quadrant information and is undefined on the imaginary axis. atan2 keeps both signs and handles axis points.
Does choosing another phase interval change the number?
No. Adding a full turn leaves cosine and sine unchanged, so the rectangular components remain the same.
What is the phase of zero?
It is undefined. Every angle with magnitude zero gives the origin, so no one angle can be recovered.